Difference equation approaches in evaluation of compound distributions.

*(English)*Zbl 0613.62020The second author [Astin Bull. 12, 22-26 (1981)], B. Sundt and W. Jewell [ibid. 12, 27-39 (1981)] and the authors [ibid. 13, 1-11 (1982)] derive computational formulae for the density of the compound distribution when the frequency distribution satisfies certain difference equations. In many instances restrictions are placed on the severity distribution.

In this paper it is shown how the results may be adapted to a wider class of severity distributions, including the inverse Gaussian and the Pareto distribution (among others). The computational techniques are clarified and in some cases simplified, and an additional class of frequency distributions is considered, which contains some other well-known distributions. It is then shown how the results may be extended to some well-known contagious frequency distributions such as the Neyman class.

In this paper it is shown how the results may be adapted to a wider class of severity distributions, including the inverse Gaussian and the Pareto distribution (among others). The computational techniques are clarified and in some cases simplified, and an additional class of frequency distributions is considered, which contains some other well-known distributions. It is then shown how the results may be extended to some well-known contagious frequency distributions such as the Neyman class.

##### MSC:

62E99 | Statistical distribution theory |

60E05 | Probability distributions: general theory |

39A10 | Additive difference equations |

##### Keywords:

compound distribution; severity distributions; inverse Gaussian; Pareto distribution; contagious frequency distributions; Neyman class
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\textit{G. E. Willmot} and \textit{H. H. Panjer}, Insur. Math. Econ. 6, 43--56 (1987; Zbl 0613.62020)

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